Semisimplicity and CM lifts

Consider the setting of a smooth variety $S$ over $\mathbb{F}_q$, and an $\ell$-adic local on $S$ which has finite determinant and is geometrically irreducible. Work of Lafforgue proves that such a local system must be pure, and it is conjectured that the action of Frobenius at closed points is semisimple. I will sketch a proof of this conjecture in the setting of mod $p$ Shimura varieties, and will deduce applications to the existence of CM lifts of certain mod p points. If time permits, I will also address the question of integral canonical models of Shimura varieties.
This is joint work with Ben Bakker and Jacob Tsimerman.